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Full text of "Mathematical And Physical Papers - Iii"

ON  THE MOTION   OF  PENDULUMS.                            83
only 0*0017 inch in defect to have existed in the measurement of the diameter.
57. I proceed next to the experiments on spheres attached to fine wires. The pendulums of this construction comprise four 1^-inch spheres, Nos. 1, 2, 3, and 4; three 2-inch spheres, Nos. 5, 6, and 7 ; and one 3-inch sphere, No. 66. Nos. 8 and 9 are the same spheres as Nos. 5 and 7 respectively, swung by suspending the wire over a cylinder instead of attaching it to a knife-edge apparatus. As this mode of suspension was not found very satisfactory, and the results are marked by Baily as doubtful cases, I shall omit the pendulums Nos. 8 and 9, more especially as with reference to the present inquiry they are merely repetitions of Nos 5 and 7.
In the case of a sphere attached to a fine wire of which the effect is neglected, and swung in an unconfmed mass of fluid, we have by the formulae (52)
2a being in this case the diameter of the sphere. Before employing this formula in the comparison of theory and experiment, it will be requisite to consider two corrections, one for the effect of the wire, the other for the effect of the confinement of the air by the sides of the vacuum tube.
I have already remarked at the end of Section IV., Part I., that the application of the formulae of Section III. to the case of such fine wires as those used in pendulum experiments is not quite safe. Be that as it may, these formulae will at any rate afford us a good estimate of the probable magnitude of the correction.
Let I be the length, a^ the radius, Y, the volume of the wire, V the volume of the sphere, / the moment of inertia of the pendulum, I' that of the air which we may conceive dragged by it, H. the sum of the elements of the mass of the pendulum multiplied by their respective vertical distances below the axis of suspension, H1 the same for the air displaced, <r the density of the air. Then the length of the isochronous simple pendulum is IH~l in vacuum, and (J+ /') (H  H')'1 in air, and the time of vibration is increased by the air in the ratio of P*H~% to
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